What are the conditions of convergence of temporal-difference learning?
There are different TD algorithms, e.g. Q-learning and SARSA, whose convergence properties have been studied separately (in many cases). 

In some convergence proofs, e.g. in the paper Convergence of Q-learning: A Simple Proof (by Francisco S. Melo), the required conditions for Q-learning to converge (in probability) are the Robbins-Monro conditions


$\sum_{t} \alpha_t(s, a) = \infty$
$\sum_{t} \alpha_t^2(s, a) &lt; \infty,$


where $\alpha_t(s, a)$ is the learning rate at time step $t$ (that can depend on the state $s$ and action $a$), and that each state is visited infinitely often. 

(The Robbins-Monro conditions (1 and 2) are due to Herbert Robbins and Sutton Monro, who started the field of stochastic approximation in the 1950s, with the paper A Stochastic Approximation Method. The fields of RL and stochastic approximation are related. See this answer for more details.)

However, note again that the specific required conditions for TD methods to converge may vary depending on the proof and the specific TD algorithm. For example, the Robbins-Monro conditions are not assumed in Learning to Predict by the Methods of Temporal Differences by Richard S. Sutton (because this is not a proof of convergence in probability but in expectation).

Moreover, note that the proofs mentioned above are only applicable to the tabular versions of Q-learning. If you use function approximation, Q-learning (and other TD algorithms) may not converge. Nevertheless, there are cases when Q-learning combined with function approximation converges. See An Analysis of Reinforcement Learning with Function Approximation by Francisco S. Melo et al. and SBEED: Convergent Reinforcement Learning with Nonlinear Function Approximation by Bo Dai et al.